Homogenization results for the calcium dynamics in living cells [610105]

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Title: Homogenization results for the calcium dynamics in living cells

Article Type: Special Issue: BIOMATH 2014

Keywords: homogenization; calcium dynamics; the bidomain model; the periodic unfolding method.

Corresponding Author: Prof. Claudia Timofte,

Corresponding Author's Institution: University of Bucharest

First Author: Claudia Timofte

Order of Authors: Claudia Timofte

Abstract: Via the periodic unfolding method, the macroscopic behavior of a nonlinear system of
coupled reaction -diffusion equations appearing in the modeling of calcium dynamics in li ving cells is
rigorously analyzed. We consider, at the microscale, two reaction -diffusion equations for the
concentrationof calcium ions in the cytosol and, respectively, in the endoplasmic reticulum, coupled
through an interfacial exchange term. Depending on the scaling of this exchange term, various models
arise at the limit. In particular, we obtain, at the macroscale, a calcium bidomain model. Such a model
is extensively used for describing the dynamics of the calcium ions, which are important intracel lular
messengers between the cytosol and the endoplasmic reticulum inside the biological cells.

Homogenization results for the calcium dynamics in
living cells
Claudia Timoftea,1
aUniversity of Bucharest, Faculty of Physics, Bucharest-Magurele, P.O. Box MG-11,
Romania
Abstract
Via the periodic unfolding method, the macroscopic behavior of a nonlinear
system of coupled reaction-diffusion equations appearing in the modeling of
calcium dynamics in living cells is rigorously analyzed. We consider, at the
microscale, two reaction-diffusion equations for the concentration of calcium ions
in the cytosol and, respectively, in the endoplasmic reticulum, coupled through
an interfacial exchange term. Depending on the scaling of this exchange term,
various models arise at the limit. In particular, we obtain, at the macroscale,
a calcium bidomain model. Such a model is extensively used for describing
the dynamics of the calcium ions, which are important intracellular messengers
between the cytosol and the endoplasmic reticulum inside the biological cells.
Keywords: homogenization, calcium dynamics, the bidomain model, the
periodic unfolding method.
2010 MSC: 35B27, 35Q92.
1. Introduction
Calcium is a very important second messenger in a living cell, participat-
ing in many cellular processes, such as protein synthesis, muscle contraction,
cell cycle, metabolism or apoptosis (see, for instance, [4]). Intracellular free
calcium concentrations must be very well regulated and many buffer proteins,
pumps or carriers of calcium take part at this complicated process. The finely
structured endoplasmic reticulum, which is surrounded by the cytosol, is an im-
portant multifunctional intracellular organelle involved in calcium homeostasis
and many of its functions depend on the calcium dynamics. The endoplasmic
reticulum plays an important role in the metabolism of human cells. It performs
diverse functions, such as protein synthesis, translocation across the membrane,
folding, etc. This complex and highly heterogeneous cellular structure spreads
throughout the cytoplasm, generating various zones with diverse morphology
Email address: [anonimizat] (Claudia Timofte)
1Tel.: +4 072 149 8466.
Preprint submitted to Elsevier November 29, 2014*Manuscript
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and functions. The study of the dynamics of calcium ions, acting as messengers
between the cytosol and the endoplasmic reticulum inside living cells, represents
a topic of huge interest, which still requires attention. Many biological mecha-
nisms involving the functions of the cytosol and of the endoplasmic reticulum
are not yet perfectly understood.
Our goal in this paper is to rigorously analyze, via the periodic unfolding
method, the macroscopic behavior of a nonlinear system of coupled reaction-
diffusion equations arising in the modeling of calcium dynamics in living cells.
We consider, at the microscale, two reaction-diffusion equations for the con-
centration of calcium ions in the cytosol and, respectively, in the endoplasmic
reticulum, coupled through an interfacial exchange term. Depending on the
scaling of this interfacial exchange term, various models arise at the limit. In
particular, we obtain, at the macroscale, a calcium bidomain model, which is
widely used for describing the dynamics of the calcium ions, acting as intracel-
lular messengers between the cytosol and the endoplasmic reticulum inside the
human cells. The calcium bidomain system consists of two reaction-diffusion
equations, one for the concentration of calcium ions in the cytosol and one for
the concentration of calcium ions in the endoplasmic reticulum, coupled through
a reaction term. For details about the physiological background of such a model,
the reader is referred to [17]. Bidomain models arise also in other contexts, such
as the modeling of diffusion processes in partially fissured media (see [3], [10]
and [11]) or the modeling of the electrical activity of the heart (see [1], [2] and
[19]).
Our models can serve as a tool for biophysicists to analyze the complex
mechanisms involved in the calcium dynamics in living cells, justifying in a
rigorous manner some biological points of view concerning such processes.
The problem of obtaining the calcium bidomain equations using homo-
genization techniques was addressed by a formal approach in [13] and by a
rigorous one, based on the use of the two-scale convergence method, in [14].
Our results constitute a generalization of some of the results contained in [13]
and [14]. The proper scaling of the interfacial exchange term has an important
influence on the limit problem and we extend the analysis from [14] to the case
in which the parameter γarising in the exchange term belongs to R. Also, the
tool we use for obtaining the above mentioned macroscopic models, namely the
periodic unfolding method, allows us to deal with a large class of heterogeneous
media.
The layout of this paper is as follows: Section 2 is devoted to the setting of
the microscopic problem. In Section 3, we present the main convergence results,
which are proven in the last section.
2. Setting of the microscopic problem
Let Ω be a bounded domain in Rn(n≥3), with a Lipschitz boundary
∂Ω consisting of a finite number of connected components. The domain Ω is
supposed to be a periodic structure made up of two connected parts, Ω"
1and
Ω"
2, separated by an interface Γ". We assume that only the phase Ω"
1reaches
2

the outer fixed boundary ∂Ω. Here,εis considered to be a small parameter
related to the characteristic size of our two regions. For modeling the dynamics
of the concentration of calcium ions in a biological cell, the phase Ω"
1represents
the cytosol, while the phase Ω"
2is the endoplasmic reticulum. Let Y1be an
open connected Lipschitz subset of the elementary cell Y= (0,1)nandY2=
Y\Y1. We consider that the boundary Γ of Y2is locally Lipschitz and that its
intersections with the boundary of Yare identically reproduced on the opposite
faces of the unit cell. Moreover, if we repeat Yin a periodic manner, the union
of all the sets Y1is a connected set, with a locally C2boundary. Also, we
consider that the origin of the coordinate system lies in a ball contained in the
above mentioned union (see [11]).
For anyε∈(0,1), let
Z"={k∈Zn|εk+εY⊆Ω},
K"={k∈Z"|εk±εei+εY⊆Ω,∀i=1,n},
whereeiare the vectors of the canonical basis of Rn. We define
Ω"
2= int(∪
k2K"(εk+εY2)),Ω"
1= Ω\Ω"
2
and we set
θ=Y\Y2.
Forα1,β1∈R, with 0< α 1< β 1, we denote by M(α1,β1,Y) the set of
all the matrices A∈(L1(Y))nnwith the following property: for any ξ∈Rn,
we have (A(y)ξ, ξ)≥α1|ξ|2,|A(y)ξ| ≤β1|ξ|, almost everywhere in Y. We
consider the matrices A"(x) =A(x/ε) defined on Ω, where A∈ M (α1,β1,Y) is
a symmetric smooth Y-periodic matrix and we denote the matrix AbyA1in
Y1and, respectively, by A2inY2.
If (0,T) is the time interval under consideration, we shall analyze the macro-
scopic behavior of the solutions of the following microscopic system:


∂u"
∂t−div (A"
1∇u") =f(u") in (0,T)×Ω"
1,
∂v"
∂t−div (A"
2∇v") =g(v") in (0,T)×Ω"
2,
A"
1∇u"·ν=A"
2∇v"·νon (0,T)×Γ",
A"
1∇u"·ν=ε
h(u",v") on (0,T)×Γ",
u"= 0 on (0 ,T)×∂Ω,
u"(0,x) =u0(x) in Ω"
1, v"(0,x) =v0(x) in Ω"
2,(2.1)
whereνis the unit outward normal to Ω"
1and the scaling exponent γis a given
real number, related to the speed of the interfacial exchange. As we shall see,
three important cases will arise at the limit, i.e. γ= 1,γ= 0 andγ=−1 (see,
also, Remark 2.). We assume that the initial conditions are non-negative and
that the functions fandgare Lipschitz-continuous, with f(0) =g(0) = 0. We
also suppose that
h(u",v") =h"
0(x)(v"−u"), (2.2)
3

whereh"
0(x) =h0(x/ε) andh0(y) is a real Y-periodic smooth function with
h0(y)≥δ>0. Besides, we consider that
H=∫
Γh0(y)dσ̸= 0.
As in [14], we can treat in a similar way the case in which the function his
given by:
h(r,s) =h(r,s)(s−r), (2.3)
with 0<hmin≤h(r,s)≤hmax<∞.
Since it is not easy to find an explicit solution of the well-posed microscopic
problem (2.1), we need to apply an homogenization procedure for obtaining a
suitable model that describes the averaged properties of the complicated mi-
crostructure. Using the periodic unfolding method proposed by D. Cioranescu,
A. Damlamian, G. Griso, P. Donato and R. Zaki (see [6] and [7]), we can find
the asymptotic behavior of the solution of our problem. For the case γ= 1, this
behavior is governed by a new nonlinear system (see (3.1)), a bidomain model .
So, in this case, at a macroscopic scale, our medium can be represented by a
continuous model, i.e. the superimposition of two interpenetrating continuous
media, the cytosol and the endoplasmic reticulum, which coexist at any point.
For the other two relevant cases, see (3.2) and (3.3).
The approach based on the periodic unfolding method allows us to work
with quite general media. For our particular geometry, we use two unfolding
operators, which map functions defined on oscillating domains into functions
defined on fixed domains and, as a consequence, we can avoid the use of exten-
sion operators and we can deal with media possessing less regularity than those
considered usually in the literature.
It might seem that these simplified assumptions about the complex calcium
dynamics inside a cell are quite strong. However, the homogenized solution fits
well with experimental data. Also, one could argue that the periodicity of the
microstructure is not a realistic assumption and it would be interesting to work
with random microstructures. Still, such a periodic structure provides a very
good description, in agreement with all the experimental findings (see [15]).
Our results hold true, also, for the case in which we assume that the ex-
terior fixed membrane ∂Ω is impermeable to calcium, i.e. we impose a homo-
geneous Neumann condition on ∂Ω. We can deal in a similar manner with
the more general case of an heterogeneous medium represented by a matrix
A"
0=A0(x,x/ε ) or by a matrix D"=D(t,x/ε ), under reasonable hypotheses
on the matrices A0andD. For example, we can assume that Dis a symmetric
matrix, with D,∂ tD∈L1(0;T;L1
per(Y))nnand such that, for any ξ∈Rn,
(D(t,x)ξ,ξ)≥α2|ξ|2,|D(t,x)ξ| ≤β2|ξ|, almost everywhere in (0 ,T)×Y, for
0<α 2<β 2.
4

3. The main convergence results
In this section, we shall present the effective behavior of the solutions of the
microscopic model (2.1) for the three important cases mentioned above.
a) Let us deal first with the case γ= 1.
Theorem 1. The solution ( u", v") of system (2.1) converges in the sense
of (4.5), as ε→0, to the unique solution ( u, v) of the following macroscopic
problem:


θ∂u
∂t−div (A1∇u)−H(v−u) =θf(u) in (0,T)×Ω,
(1−θ)∂v
∂t−div (A2∇v) +H(v−u) = (1 −θ)g(v) in (0,T)×Ω,
u(0,x) =u0(x), v(0,x) =v0(x) in (0,T)×Ω.(3.1)
Here,A1andA2are the homogenized matrices, given by:
A1
ij=∫
Y1(
aij+aik∂χ1j
∂yk)
dy,
A2
ij=∫
Y2(
aij+aik∂χ2j
∂yk)
dy
andχ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n , are the weak solutions
of the cell problems


−∇y·((A1(y)∇yχ1k) =∇yA1(y)ek, y∈Y1,
(A1(y)∇yχ1k)·ν=−A1(y)ek·ν, y ∈Γ,


−∇y·((A2(y)∇yχ2k) =∇yA2(y)ek, y∈Y2,
(A2(y)∇yχ2k)·ν=−A2(y)ek·ν, y ∈Γ.
At a macroscopic scale, we obtain a continuous model, a so-called bidomain
model , similar to those arising in the context of the modeling of diffusion pro-
cesses in partially fissured media (see [3] and [11]) or in the case of the modeling
of the electrical activity of the heart (see [1], [2] and [19]). If we assume that h
is given by (2.3), then, at the limit, the exchange term appearing in (3.1) is of
the form |Γ|h(u,v).
b) Forγ= 0, i.e. for high contact resistance, we obtain, at the macroscale,
only one concentration field. So, u=v=u0andu0is the unique solution of
the following problem:


∂u0
∂t−div (A0∇u0) =θf(u0) + (1 −θ)g(u0) in (0,T)×Ω,
u0(0,x) =u0(x) +v0(x) in (0,T)×Ω.(3.2)
5

Here, the effective matrix A0is given by:
A0
ij=∫
Y1(
aij+aik∂χ1j
∂yk)
dy+∫
Y2(
aij+aik∂χ2j
∂yk)
dy,
in terms of the functions χ1k∈H1
per(Y1)/R, χ2k∈H1
per(Y2)/R,k= 1,…,n,
weak solutions of the local problems


−∇y·((A1(y)∇yχ1k) =∇yA1(y)ek, y∈Y1,
(A1(y)∇yχ1k)·ν=−A1(y)ek·ν, y ∈Γ,


−∇y·((A2(y)∇yχ2k) =∇yA2(y)ek, y∈Y2,
(A2(y)∇yχ2k)·ν=−A2(y)ek·ν, y ∈Γ.
In this case, the exchange at the interface leads to the modification of the
limiting diffusion matrix, but the insulation is not enough strong to impose the
existence of two different limit concentrations.
c) For the case γ=−1, i.e. for very fast interfacial exchange of calcium
between the cytosol and the endoplasmic reticulum (i.e. for weak contact resis-
tance), at the limit, we also obtain u=v=u0and, in this case, the effective
concentration field u0satisfies:


∂u0
∂t−div (A0∇u0) =θf(u0) + (1 −θ)g(u0) in (0,T)×Ω,
u0(0,x) =u0(x) +v0(x) in (0,T)×Ω.(3.3)
The effective coefficients are given by:
A0;ij=∫
Y1(
aij+aik∂w1j
∂yk)
dy+∫
Y2(
aij+aik∂w2j
∂yk)
dy,
wherew1k∈H1
per(Y1)/R, w2k∈H1
per(Y2)/R,k= 1,…,n, are the weak solu-
tions of the cell problems


−∇y·(A1(y)∇yw1k) =∇yA1(y)ek, y∈Y1,
−∇y·(A2(y)∇yw2k) =∇yA2(y)ek, y∈Y2,
(A1(y)∇yw1k)·ν= (A2(y)∇yw2k)·ν y∈Γ,
(A1(y)∇yw1k)·ν+h0(y)(w1k−w2k) =−A1(y)ek·ν y∈Γ.
It is important to notice that the diffusion coefficients depend now on h0.
Remark 2. For simplicity, we address here only the relevant cases γ=
−1,0,1. For the case γ∈(−1,1), we get, at the limit, the macroscopic problem
(3.2), while for γ > 1, we obtain a problem similar to (3.1), but without the
exchange terms. Finally, for the case γ <−1, we obtain, at the limit, a standard
composite medium without any barrier resistance.
6

Let us notice that the geometry of the domain Ω is crucial. For instance,
as proven in a counterexample of H.K. Hummel [16], if both phases are dis-
connected, for γ > 1, the sequence of solutions of the microscopic problem
might diverge. Also, if we deal with a different geometry, i.e. if we suppose
that the phase Ω"
1is still connected, but Ω"
2is disconnected, it follows that the
homogenized matrix A2= 0 and, for γ= 1, the system (3.1) consists of two
coupled equations, a partial differential equation and an ordinary differential
one, which, in particular cases, can be solved, leading us to only one partial
differential equation with memory.
Remark 3. The conditions imposed on the nonlinear functions f,gand
hcan be relaxed. For instance, we can consider that fandgare maximal
monotone graphs, verifying suitable growth conditions (see [8]). Also, as in [18],
[20] and [23], we can work with more general functions h.
Remark 4. Let us notice that we can also treat the case in which the initial
conditions depend on ε. A standard choice is to consider that u"(0,x) =u"
0(x)
in Ω"
1,v"(0,x) =v"
0(x) in Ω"
2, withu"
0(x)∈L2(Ω"
1), v"
0(x)∈L2(Ω"
2). We assume
that the extensions by zero of these functions verify eu"
0⇀u 0,ev"
0⇀v 0, weakly
inL2(Ω). In this situation, for γ= 1 (see Theorem 1.), we get u(0,x) =θu0(x)
andv(0,x) = (1 −θ)v0(x), while for the other cases, namely b) and c), we obtain
u(0,x) =θu0(x) + (1 −θ)v0(x) andu0(0,x) =θu0(x) + (1 −θ)v0(x). .
4. Proof of the main results
Let us introduce now the function spaces and norms we shall work with in
the sequel. Let
H1
@Ω(Ω"
1) ={v∈H1(Ω"
1)|v= 0 on∂Ω∩∂Ω"
1},
V(Ω"
1) =L2(0,T;H1
@Ω(Ω"
1)),V(Ω"
1) ={v∈V(Ω"
1)|∂tv∈L2((0,T)×Ω"
1)},
V(Ω"
2) =L2(0,T;H1(Ω"
2)),V(Ω"
2) ={v∈V(Ω"
2)|∂tv∈L2((0,T)×Ω"
2)},
with
(u(t),v(t))Ω"
=∫
Ω"
u(t,x)v(t,x)dx,∥u(t)∥2
Ω"
= (u(t),u(t))Ω"
,
(u,v)Ω";t=t∫
0(u(t),v(t))Ω"dt,∥u∥2
Ω";t= (u,u)Ω";t,
forα= 1,2. Also, let
V(Ω) =L2(0,T;H1(Ω)),V(Ω) = {v∈V(Ω)|∂tv∈L2((0,T)×Ω)},
with
(u(t),v(t))Ω=∫
Ωu(t,x)v(t,x)dx,∥u(t)∥2
Ω= (u(t),u(t))Ω,
7

(u,v)Ω;t=t∫
0(u(t),v(t))Ωdt,∥u∥2
Ω;t= (u,u)Ω;t
and
V0(Ω) = {v∈V(Ω)|v= 0 on∂Ω a.e. on (0 ,T)},V0(Ω) =V0(Ω)∩ V(Ω).
The variational formulation of problem (2.1) is as follows: find ( u",v")∈
V(Ω"
1)× V(Ω"
2), with (u"(0,x),v"(0,x)) = (u0(x),v0(x))∈(L2(Ω))2and
(∂u"(t),φ(t))Ω"
1+ (∂v"(t),ψ(t))Ω"
2+
(A"
1(t)∇u",∇φ(t))Ω"
1+ (A"
2(t)∇v",∇ψ(t))Ω"
2−
ε
(h(u",v"),φ(t)−ψ(t))Γ"= (f(u"(t)),φ(t))Ω"
1+ (g(v"(t)),ψ(t))Ω"
2,(4.1)
for a.e.t∈(0,T) and any ( φ,ψ)∈V(Ω"
1)×V(Ω"
2).
Following the same techniques used in [14], it is not difficult to prove that
(4.1) is a well-posed problem and that u"andv"are non-negative and bounded
almost everywhere.
Taking (u",v") as test function in (4.1), integrating with respect to time and
taking into account that u"andv"are bounded and non-negative, it follows that
there exists a constant C≥0, independent of ε, such that
∥u"(t)∥2
Ω"
1+∥v"(t)∥2
Ω"
2+∥∇u"∥2
Ω"
1;t+∥∇v"∥2
Ω"
2;t+
ε
(h(u",v"),u"−v")Γ";t≤C, (4.2)
for a.e.t∈(0,T). Also, as in [14] or [23], we can see that there exists a positive
constantC≥0, independent of ε, such that
∥∂tu"(t)∥2
Ω"
1+∥∂tv"(t)∥2
Ω"
2≤C, (4.3)
forγ≥1 and
∥∂tu"∥L2(0;T;H1(Ω"
1))+∥∂tv"∥L2(0;T;H1(Ω"
2))≤C, (4.4)
forγ <1. These a priori estimates will allow us to use the periodic unfolding
method and to obtain the needed convergence results in all the above men-
tioned relevant cases. For retrieving the macroscopic behavior of the solution
of problem (2.1), we use two unfolding operators, T"
1andT"
2, which transform
functions defined on oscillating domains into functions defined on fixed domains
(see [5], [6] and [9]).
Let us prove now Theorem 1. For γ= 1, using the obtained a priori es-
timates and the properties of the operators T"
1andT"
2, it follows that there
existu∈L2(0,T;H1
0(Ω)),v∈L2(0,T;H1(Ω)),bu∈L2((0,T)×Ω;H1
per(Y1)),
bv∈L2((0,T)×Ω;H1
per(Y2)) such that, passing to a subsequence, for ε→0, we
have:

T"
1(u")→ustrongly in L2((0,T)×Ω,H1(Y1)),
T"
1(∇u")⇀∇u+∇ybuweakly inL2((0,T)×Ω×Y1),
T"
2(v")⇀v weakly inL2((0,T)×Ω,H1(Y2)),
T"
2(∇v")⇀∇v+∇ybvweakly inL2((0,T)×Ω×Y2).(4.5)
8

Moreover, as in [14] and [21],∂u
∂t∈L2(0,T;L2(Ω)),∂v
∂t∈L2(0,T;L2(Ω)) and
u∈C0([0,T];H1
0(Ω)), v∈C0([0,T];H1(Ω)). So,u∈ V 0(Ω) andv∈ V(Ω).
Let us mention that, in fact, under our hypotheses, passing to a subsequence,
T"
1(u") converges strongly to uinLp((0,T)×Ω×Y1), for 1 ≤p <∞. As
a consequence, since the Nemytskii operator corresponding to the nonlinear
functionfis continuous, it follows that f(T"
1(u")) converges to f(u). A similar
result holds true for v".
For getting the limit problem (3.1), we take, as test functions,
φ(t,x) =φ1(t,x) +εφ2(t,x,x
ε),
ψ(t,x) =ψ1(t,x) +εψ2(t,x,x
ε),
withφ1,ψ1∈C1
0((0,T)×Ω),φ2∈C1
0((0,T)×Ω;H1
per(Y1)) andψ2∈
C1
0((0,T)×Ω;H1
per(Y2)).
We have:
∫T
0∫
Ω"
1∂u"
∂tφdxdt +∫T
0∫
Ω"
2∂v"
∂tψdxdt +∫T
0∫
Ω"
1A"
1∇u"· ∇φdxdt +
∫T
0∫
Ω"
2A"
2∇v"· ∇ψdxdt −ε∫T
0∫
Γ"h(u",v")(φ−ψ)dσdt =
∫T
0∫
Ω"
1f(u")φdxdt +∫T
0∫
Ω"
2g(v")ψdxdt. (4.6)
Applying the corresponding unfolding operators in (4.6), we obtain:
∫T
0∫
ΩY1∂T"
1(u")
∂tT"
1(φ)dxdydt +∫T
0∫
ΩY2∂T"
2(v")
∂tT"
2(ψ)dxdydt +
∫T
0∫
ΩY1T"
1(A"
1)T"
1(∇u")· T"
1(∇φ)dxdydt +
∫T
0∫
ΩY2T"
2(A"
2)T"
2(∇v")· T"
2(∇ψ)dxdydt +
∫T
0∫
ΩΓh0(y)(T"
1(u")− T"
2(v"))(T"
1(φ)− T"
2(ψ))dxdσdt =
∫T
0∫
ΩY1T"
1(f(u"))T"
1(φ)dxdydt +∫T
0∫
ΩY2T"
2(g(v"))T"
2(ψ)dxdydt. (4.7)
Using the above convergence results and Lebesgue’s convergence theorem, we
can pass to the limit in (4.7) (see, for details, [6], [8], [21] and [22]).
9

Thus, we get:
∫T
0∫
ΩY1∂tu(t,x)φ1(t,x)dxdt +∫T
0∫
ΩY2∂tv(t,x)ψ1(t,x)dxdt+
∫T
0∫
ΩY1A1(y)(∇u+∇ybu)·(∇xφ1+∇yφ2)dxdydt +
∫T
0∫
ΩY2A2(∇v+∇ybv)·(∇xψ1+∇yψ2)dxdydt +
∫T
0∫
ΩΓh0(y)(u−v)(φ1−ψ1)dxdσdt =
∫T
0∫
ΩY1f(u)φ1dxdydt +∫T
0∫
ΩY2g(v)ψ1dxdydt. (4.8)
Using standard density arguments, it follows that (4.8) holds true for any φ1∈
L2(0,T;H1
0(Ω)),ψ1∈L2(0,T;H1(Ω)),φ2∈L2((0,T)×Ω;H1
per(Y1)) andψ2∈
L2((0,T)×Ω;H1
per(Y2)). This is just the weak formulation of the limit problem
(3.1). Indeed, if we recall that θ=|Y1|, 1−θ=|Y2|and we take φ1= 0 and,
respectively, ψ1= 0, using the local problems, we obtain exactly (3.1). We
notice that passing to the limit, with ε→0, in the initial conditions, we get
u(0,x) =u0(x), v(0,x) =v0(x), for allx∈Ω.
Sinceuandvare uniquely determined (see [14]), the above convergences for
the microscopic solutions hold for the whole sequence and this ends the proof
of Theorem 1.
Let us treat now the other two relevant cases. We observe that
∥T"
1(u")− T"
2(v")∥L2((0;T)ΩΓ)≤Cε1

2.
Thus, for the case γ= 0, we have, at the macroscale,
u=v=u0∈ V 0(Ω)
and, by unfolding and passing to the limit, we get
∫T
0∫
ΩY1∂tu0(t,x)ϕ(t,x)dxdt +∫T
0∫
ΩY2∂tu0(t,x)ϕ(t,x)dxdt+
T∫
0∫
ΩY1A1(∇u0+∇ybu)·(∇xΦ +∇yeφ1)dxdy +
T∫
0∫
ΩY2A2(∇u0+∇ybv)·(∇xΦ +∇yeφ2)dxdy =
10

T∫
0∫
ΩY1f(u0)Φdxdy +T∫
0∫
ΩY2g(u0)Φdxdy,
for Φ ∈L2(0,T;H1
0(Ω)),eφ1∈L2((0,T)×Ω;H1
per(Y1)) and eφ2∈L2((0,T)×
Ω;H1
per(Y2)), which leads immediately to the macroscopic problem (3.2).
For the case γ=−1, we still obtain at the macroscale
u=v=u0∈ V 0(Ω).
Moreover, in this case, following the techniques from [9], one can prove that
T"
1(u")− T"
2(v")
ε⇀bu−bvweakly in L2((0,T)×Ω×Γ).
Hence, by unfolding and passing to the limit, we get
∫T
0∫
ΩY1∂tu0(t,x)ϕ(t,x)dxdt +∫T
0∫
ΩY2∂tu0(t,x)ϕ(t,x)dxdt+
T∫
0∫
ΩY1A1(∇u0+∇ybu)·(∇xΦ +∇yeφ1)dxdy +
T∫
0∫
ΩY2A2(∇u0+∇ybv)·(∇xΦ +∇yeφ2)dxdy +
T∫
0∫
ΩΓh0(bu−bv)(eφ1−eφ2)dxdσ =T∫
0∫
ΩY1f(u0)Φdxdy +T∫
0∫
ΩY2g(u0)Φdxdy,
for Φ ∈L2(0,T;H1
0(Ω)),eφ1∈L2((0,T)×Ω;H1
per(Y1)) and eφ2∈L2((0,T)×
Ω;H1
per(Y2)), which leads to the limit problem (3.3).
5. Conclusions
Via the periodic unfolding method, the asymptotic behavior of the solution
of a system of coupled partial differential equations arising in the modeling of
calcium dynamics in living cells was analyzed. At the microscale, two reaction-
diffusion equations for the concentration of calcium ions in the cytosol and,
respectively, in the endoplasmic reticulum, coupled through an interfacial ex-
change term, were considered. Depending on the scaling of the exchange term,
three important cases were considered. The advantage provided by our approach
is that, avoiding the use of extension operators, it allows us to deal with quite
general media.
11

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